Tenth Workshop on NonPerturbative QCD lInstitut dAstrophysique de Paris Paris 11 June 2009 Brownian Motion in AdSCFT J de Boer V E Hubeny M Rangamani MS Brownian motion in AdSCFT arXiv08125112 ID: 673805
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Slide1
Masaki ShigemoriUniversity of AmsterdamTenth Workshop on Non-Perturbative QCDl’Institut d’Astrophysique de ParisParis, 11 June 2009
Brownian Motion in AdS/CFTSlide2
J. de Boer, V. E. Hubeny, M. Rangamani, M.S., “Brownian motion in AdS/CFT,” arXiv:0812.5112.A. Atmaja, J. de Boer, K. Schalm, M.S., work in progress.2This talk is based on:Slide3
3Intro / MotivationSlide4
4
AdS/CFT and fluid-gravity
AdS
CFT
black hole in
quantum gravity
horizon dynamics
in classical GR
plasma in strongly
coupled QFT
hydrodynamics
Navier-Stokes eq.
Long-wavelength approximation
difficult
easier;
better-understood
Bhattacharyya+Minwalla
+
Rangamani+Hubeny
0712.2456
?Slide5
Hydro: coarse-grained5Macrophysics vs. microphysics
BH in c
lassical GR is also macro, approx. description
of underlying microphysics of QG BH!
Can’t study microphysics within hydro framework
(by definition)
want to go beyond hydro approx
coarse
grainSlide6
― Historically, a crucial step toward microphysics of nature1827 BrownDue to collisions with fluid particlesAllowed to determine Avogadro #:UbiquitousLangevin eq. (friction + random force)
6
Brownian motion
Robert Brown (1773-1858)
erratic motion
pollen particleSlide7
Do the same for hydro. in AdS/CFT!Learn about QG from BM on boundaryHow does Langevin dynamics come aboutfrom bulk viewpoint?Fluctuation-dissipation theoremRelation to RHIC physics?
7
Brownian motion in AdS/CFT
Related work:
drag force:
Herzog+Karch+Kovtun+Kozcaz+Yaffe
,
Gubser
,
Casalderrey-Solana+Teaney
transverse momentum broadening:
Gubser
,
Casalderrey-Solana+TeaneySlide8
8Preview: BM in AdS/CFT
horizon
AdS boundary
at infinity
fundamental
string
black hole
endpoint =
Brownian particle
Brownian motionSlide9
Intro/motivationBMBM in AdS/CFTTime scalesBM on stretched horizon
9
Outline Slide10
10Brownian motion
Paul Langevin (1872-1946)Slide11
11Langevin dynamics
Generalized Langevin
eq
:
delayed friction
random forceSlide12
12General properties of BM
Displacement:
diffusive regime
(random walk)
ballistic regime
(
init.
velocity )
diffusion constantSlide13
13Time scales
Relaxation time
Collision duration time
Mean-
free-path time
time elapsed
in a single collision
Typically
but not necessarily so
for strongly coupled plasma
R
(
t
)
t
time between collisionsSlide14
14BM in AdS/CFTSlide15
AdS Schwarzschild BH15Bulk BM
horizon
AdS boundary
at infinity
fundamental
string
black hole
endpoint =
Brownian particle
Brownian motion
rSlide16
Horizon kicks endpoint on horizon(= Hawking radiation)Fluctuation propagates toAdS boundaryEndpoint on boundary
(= Brownian particle) exhibits BM
16
Physics of BM in AdS/CFT
horizon
boundary
endpoint =
Brownian particle
Brownian motion
r
transverse fluctuation
kick
Whole process is dual to quark hit by QGP particlesSlide17
17BM in AdS/CFT
horizon
boundary
r
Probe approximation
Small
g
s
No
interaction with bulk
The only
interaction
is
at horizon
Small fluctuation
Expand Nambu-
Goto
action
to quadratic order
Transverse positions
are similar to Klein-Gordon scalarsSlide18
Quadratic action18Brownian string
d=3: can be solved exactly
d>3: can be solved in low frequency limit
Mode expansionSlide19
Near horizon:
19
Bulk-boundary dictionary
outgoing
mode
ingoing
mode
phase shift
: tortoise coordinate
: cutoff
observe BM
in gauge theory
correlator of
radiation modes
Can learn about quantum gravity in principle!
Near boundarySlide20
Semiclassically, NH modes are thermally excited:20Semiclassical analysis
Can use dictionary to compute
x
(
t
),
s
2
(
t
)
(bulk
boundary)
ballistic
diffusive
Does exhibit
Brownian motionSlide21
21Time scalesSlide22
22
Time scales
R
(
t
)
t
information about plasma constituentsSlide23
23Time scales from R-correlatorsSimplifying assumptions:
: shape of a single
p
ulse
: random sign
: number of pulses per unit time,
R
(
t
) : consists of many pulses randomly distributed
Distribution of pulses = Poisson distributionSlide24
24Time scales from R-correlatorsCan determine μ, thus t
mfp
tilde = Fourier transform
2
-pt
func
Low-freq.
4
-pt
funcSlide25
25Sketch of derivationProbability that there are
k
pulses in period
[0,
τ
]:
0
k
pulses
…
(Poisson dist.)
2-pt
func
:Slide26
26
Sketch of derivation
Similarly, for 4-pt
func
,
“disconnected part”
“connected
part”Slide27
27R-correlators from BM in AdS/CFT
Expansion of NG action to higher order:
Can compute
t
mfp
from correction to 4-pt
func
.
Can compute
and thus
t
mfpSlide28
28
Times scales from AdS/CFT
conventional k
inetic
theory is good
Resulting timescales:
weak couplingSlide29
29
Times scales from AdS/CFT
Multiple collisions occur simultaneously.
Resulting timescales:
strong coupling
is also possible.
.
Cf. “fast scrambler”Slide30
30BM on stretched horizon (Jorge’s talk)Slide31
31ConclusionsSlide32
Boundary BM ↔ bulk “Brownian string” can study QG in principleSemiclassically, can reproduce Langevin
dyn
. from bulk
random force
↔
Hawking rad. (kicking by horizon)
friction
↔
absorption
Time scales in strong coupling QGP:
BM on stretched horizon (Jorge’s talk
)
Fluctuation-dissipation theorem
32Conclusions Slide33
33Thanks!